\section{Vectors}

\definition{}
Elements of $\mathbb{R}^n$ are often called \textit{vectors}. \\
As you may already know, we have a few operations on vectors:
\begin{itemize}
	\item Vector addition: $[a_1, a_2] + [b_1, b_2] = [a_1+b_1, a_2+b_2]$
	\item Scalar multiplication: $x \times [a_1, a_2] = [xa_1, xa_2]$.
\end{itemize}
\note{
	The above examples are for $\mathbb{R}^2$, and each vector thus has two components. \\
	These operations are similar for all other $n$.
}

\problem{}
Compute the following or explain why you can't:
\begin{itemize}
	\item $[1, 2, 3] - [1, 3, 4]$ \note{Subtraction works just like addition.}
	\item $4 \times [5, 2, 4]$
	\item $a + b$, where $a \in \mathbb{R}
^5$ and $b \in \mathbb{R}^7$
\end{itemize}

\vfill

\problem{}
Consider $(2, -1)$ and $(3, 1)$ in $\mathbb{R}^2$. \\
Can you develop geometric intuition for their sum and difference?

\begin{center}
	\begin{tikzpicture}[scale=1]

		\draw[->]
			(0,0) coordinate (o) -- node[below left] {$(2, -1)$}
			(2, -1) coordinate (a)
		;

		\draw[->]
			(a) -- node[below right] {$(3, 1)$}
			(5, 0) coordinate (b)
		;

		\draw[
			draw = gray,
			text = gray,
			->
		]
			(o) -- node[above] {$??$}
			(b) coordinate (s)
		;

	\end{tikzpicture}
	\end{center}


\vfill
\pagebreak

\definition{Euclidean Norm}
A \textit{norm} on $\mathbb{R}^n$ is a map from $\mathbb{R}^n$ to $\mathbb{R}^+_0$ \\
Usually, one thinks of a norm as a way of mesuring \say{length} in a vector space. \\
The norm of a vector $v$ is written $||v||$. \\

\vspace{2mm}

We usually use the \textit{Euclidean norm} when we work in $\mathbb{R}^n$. \\
If $v \in \mathbb{R}^n$, the Euclidean norm is defined as follows: \\
If $v = [v_1, v_2, ..., v_n]$,
$$
	||v|| = \sqrt{v_1^2 + v_2^2 + ... + v_n^2}
$$
This is simply an application of the Pythagorean theorem.

\problem{}
Compute the euclidean norm of
\begin{itemize}
	\item $[2, 3]$
	\item $[-2, 1, -4, 2]$
\end{itemize}

\vfill

\problem{}
Show that $a \cdot a$ is $||a||^2$.

\vfill

\pagebreak